To determine which of the given functions are exponential decay functions, we need to understand the general form of an exponential decay function. An exponential decay function typically has the form [tex]\( f(x) = a \cdot b^x \)[/tex] where:
- [tex]\( a \)[/tex] is a constant.
- [tex]\( b \)[/tex] is a positive base less than 1, i.e., [tex]\( 0 < b < 1 \)[/tex].
- [tex]\( x \)[/tex] is the exponent.
Let's analyze each function to see which ones fit this form.
### Analysis of Each Function
1. [tex]\( f(x) = \frac{3}{4}\left(\frac{7}{4}\right)^x \)[/tex]
Here, the base [tex]\( \frac{7}{4} \)[/tex] is greater than 1. Since [tex]\( b > 1 \)[/tex], this function represents exponential growth, not decay.
2. [tex]\( f(x) = \frac{2}{3}\left(\frac{4}{5}\right)^{-x} \)[/tex]
The base in this function is [tex]\( \frac{4}{5} \)[/tex], which is between 0 and 1. However, the exponent is [tex]\( -x \)[/tex]. We can rewrite the function to better understand its behavior:
[tex]\[
f(x) = \frac{2}{3} \left(\frac{4}{5}\right)^{-x} = \frac{2}{3} \cdot \left(\frac{5}{4}\right)^x
\][/tex]
Simplifying, we see the base of the exponent [tex]\( \left(\frac{5}{4}\right) > 1 \)[/tex]. When raising to [tex]\( x \)[/tex], it effectively becomes less than 1 due to the negative exponent making it [tex]\( \frac{4}{5} \)[/tex]. Thus, this function represents exponential decay.
3. [tex]\( f(x) = \frac{3}{2}\left(\frac{8}{7}\right)^{-x} \)[/tex]
Similar to the previous function, the base here is [tex]\( \frac{8}{7} \)[/tex]. With the negative exponent [tex]\( -x \)[/tex], we can rewrite it as:
[tex]\[
f(x) = \frac{3}{2} \left(\frac{8}{7}\right)^{-x} = \frac{3}{2} \cdot \left(\frac{7}{8}\right)^x
\][/tex]
Simplifying, the base [tex]\( \left(\frac{7}{8}\right) < 1 \)[/tex], indicating an exponential decay function.
4. [tex]\( f(x) = \frac{1}{3}\left(-\frac{9}{2}\right)^x \)[/tex]
The base here is [tex]\( -\frac{9}{2} \)[/tex], which is negative. Exponential decay functions involve positive bases that are less than 1. A negative base does not fit the typical exponential decay model, thus this is not considered an exponential decay function.
### Conclusion
After analyzing each function, we find that the second and third functions are exponential decay functions. Therefore, the correct answer is:
[tex]\[ (2, 3) \][/tex]
